All right, so that's our first linear function.

Let's have a look at a second linear function.

Again, talk about interpretation of coefficients.

So here, I'm thinking about a production process, and

I'm interested in modeling the time to produce as a function

of the number or the quantity of units that I'm producing.

So obviously, such a function would be very helpful if you had a customer who

gave you an order, one of the first things a customer is going to say to you is when

is it going to be ready?

Well, how long does it take to produce?

That's the idea here.

And so, it certainly answers some practical questions,

the time to produce function.

So in the example that I'm looking at, we're given some information.

The information as it takes two hours to set up a production run.

And each incremental unit produced, every extra unit,

always takes an additional 15 minutes.

15 minutes is a quarter, 0.25, of an hour.

Now, in terms of modeling this, there's a key word here, and that's the word always.

And what that is telling you is that the time to produce goes up by 15 minutes,

regardless of the number of units being produced.

So that's the constant slope statement coming in that is associated

with the linear function or straight line function.

So it's that always there that it's telling me that we're looking at

a straight line function.

So if we were to write down these words in terms of a quantitative model,

then we need to start defining variables.

So let's call T, the time to produce q units.

Then, what we're told is that the time to produce q units always

starts off with two hours.

There's a two-hour setup time.

And then, once we've set the machine up, it's quarter of an hour,

0.25 of an hour to produce each additional unit.

And so in this example, the interpretation of b is the setup time,

and m, I might call the work rate, which is 15 minutes per additional item.

I certainly like to use the word rate here when we're talking about a slope,

because a slope is a rate of change.

And so in this example, we were given the words associated with the process,

and it's really up to us to turn it into a mathematical or modelling formulation.

So the first bullet point is the description of the process.

The second bullet point is the articulation of the process

in terms of a quantitative model.

So there's a second example.

So once again, we've got interpretations in the first example where we had

the linear cost function.

Our intercept and slope were fixed and variable cost.

This time around in the time-to-produce function,

they are setup time and, as I've termed it here, the work rate.

So, with this function at hand, I'm going to be able to predict how long

it takes to produce a job of any particular size.

And so, let's just check out the graph here quickly.

We should confirm by looking at the axis, and once again,

we've got the input to the model, that's the quantity on the x-axis,

and the output, the time to produce, on the y-axis.

We've called them T and q here.

We look at the line and look to see where it intercepts the point X equal to zero.

By just looking at the scale, we can say, yes, that's about two.

And we could confirm for ourselves, for example, by looking to see how much

the graph goes up between 20 and 30, that's a 10 unit change in X.

For 10 unit change in x, we're getting 2.5 extra hours to produce.

So I'm just eyeballing this graph to confirm that it is consistent with

the equation that I've written down.

And it's always a good idea to do that, because mistakes happen.

And it's good to have in place some kind of checks as we go along the way.

So there's our equation and the graphical representation of it, so

a model for a time-to-produce.

I want to briefly talk about a topic that uses

linear functions as an essential input.

Now, in this particular course, I'm not going to show you the implementation, but

I just want you to know that this technique is out there.

It solves a set of problems, and it is totally focused on linear functions.

And that technique is known as linear programming.

It's one of the work horses of operations research,

it often goes by the acronym LP.

And it is used to solve a certain set of optimization problems.

And those are optimization problems where all the features of the underlying

process can be captured in linear, with a linear construct, basically lots of lines.

One of the interesting things about these linear programs is that

they explicitly incorporate what we term as constraints.

So when we try to optimize processes that really means doing the best that we can,

it's often important to recognize that we work within constraints.

So there's no point coming up with an optimal solution that we can't achieve,

because we don't have enough workers, or

we don't have enough of a certain product on hand to achieve that optimization.

And so, constraints are ideas that we can incorporate

in our modeling process to try and make sure that our models really do

correspond to the world that we're trying to describe.

And as I say linear programming really does think carefully about

incorporating those constraints.

They just happen to be linear constraints in linear programming.

So, if you come across problems that are to do with optimization, and

most of all of the underlying features of the process can be captured through

a linear representation, then linear programming might be the thing for you.

And you can often find linear programming implemented in

spreadsheets sometimes with add-ins.

And so, Excel has a solver which can be used for doing linear programming.

So this is one of the big uses of linear models for optimization.

Again, it's not a part of this particular course, but I want you to know that it's

out there, and it's one of the, as I say, big uses of linear models.